Crabs might be another example (10 limbs but only 8 legs with common sideways--but not 360 degree--motion).
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Paul A. ClaytonJan 28 '13 at 22:46

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@Kevin, "octopuses" is better form although octopi is used under the mistaken assumption that the word is latin. If you want to be pedantic, it should be octopodes or octopoda since it comes from Greek. See here for a nice discussion.
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terdonJan 28 '13 at 23:36

2 Answers
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No, there is nothing special about eight legs. You have species with two legs (humans), four legs (most mammals), six legs (insects), eight legs (spiders), ten legs (lobsters), and many more legs (millipedes, centipedes).

@PaulA.Clayton I am not sure what it means to have 360 deg. motion. All organisms use 360 degree motion. One could argue that radially symmetrical organisms lack a preferred "head" direction, but spiders and octopuses are not radially symmetrical. Why would a spider be more 360 degree motive than a crab, a mouse, or an insect?
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KennyPeanutsJan 29 '13 at 1:45

@KennyPeanuts, I think both the OP and I, in my previous comment, were laboring under a misapprehension. 180 degree vision and motion would be more correct. A spider has eyes situated all around its head, allowing for 180 degree vision. The lack of a specific "forward" direction in this species would indeed indicate a greater ability for 180 degree movement. Not so octopuses though, I agree.
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terdonJan 29 '13 at 20:17

I suppose that octopodes (and really all animals, but especially those that swim and fly) have 41,253°² motion, not 360° motion, because they exist in a three-dimensional world.
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OreotrephesJul 28 '13 at 14:28

This is just a wild guess, but for movement on a surface in which legs are lifted from the surface, six legs might be sufficient to provide three points of support at all times (think three-legged stool) while allowing relatively rapid movement (legs on both sides can be moved simultaneously), but such might be both less stable and less friendly to variable direction of movement (since legs are in fixed locations and not evenly/radially distributed in animals with bilateral symmetry, having more legs would facilitate balance for different directions of movement). Having eight legs allows well distributed support and might allow more simplified motor control.

Having more than eight legs would tend to lengthen the body (for animals with bilateral symmetry) which might tend to bias movement direction (as well as possibly being wasteful of resources).

There might also be correlation of highly variable movement direction with requirements for greater stabilization. E.g., eight legs might provide greater stability on a web (which surface might encourage more variable movement directions) and better support large claws (which might make sideways movement more useful in face-to-face encounters).

Note that the above does not address non-walking animals nor animals with radial symmetry (For an animal with radial symmetry, six legs might be sufficient to provide support with three.).

Hopefully, a better answer will be given, since I have not even paid attention to how spiders (or crabs) move. (The manner of movement would give insight into how significant support/stability and motor control are.)